Quantum Computing
(Fall 2013)
Instructor: Shengyu Zhang.
Lecture 9 Quantum Information 1: operations and distance
In this lecture, we give some background knowledge which is necessary for the future lectures on
quantum information theory.
1. Quantum operations
We’ve encountered some quantum operations, including unitary operation and orthogonal
measurements. One can of course have other operations, such as adding a quantum system
and discarding part of a system. In general, one can use arbitrary sequence of the above
operations to an existing system A, such as attaching a system B, applying a unitary U on AB,
removing A, and measure B. This is physically clear, but mathematically not friendly to
process. Is there a characterization for all these implementable quantum operations? It turns
out to be yes, though the answer is slightly complicated. We will mention two most
commonly used representations, one is operator-sum, and another is CPTP maps.
Recall that a general (possibly mixed) quantum state in space of dimension 𝑑 is a square
matrix 𝜌 with 𝑡𝑟(𝜌) = 1 and 𝜌 ≽ 0.
Suppose that a joint system (A,B) is in a quantum state 𝜌𝐴𝐵 . Then removing system B means
to trace out B and get 𝑡𝑟𝐵 (𝜌𝐴𝐵 ).
Suppose that system A is in state 𝜌𝐴 and system B is in state 𝜌𝐵 which is not entangled
with system A. Adding B to A results in the state 𝜌𝐴 ⊗ 𝜌𝐵 .
1.1. operator-sum
For the first one, sometimes also called Kraus representation, let’s first examine a closely
related operation as an easy start. We’ve learned orthogonal measurements, but we can use
adding/removing systems to get more measurements. For example, first add a system, then
make an orthogonal measurement, and finally discard some subsystem. What’s the resulting
measurement? It turns out that all such measurements can be described by the so-called
POVM measurements. It is just a collection of operators {𝑀𝑖 } s.t. ∑𝑖 𝑀𝑖† 𝑀𝑖 = 𝐼. When we
use this measurement on a quantum state 𝜌, we observe outcome 𝑖 with probability 𝑝𝑖 =
𝑡𝑟(𝑀𝑖 𝜌𝑀𝑖† ) and the post-measurement state becomes 𝜌𝑖 =
†
𝑀𝑖 𝜌𝑀𝑖
†
. The
𝑡𝑟(𝑀𝑖 𝜌𝑀𝑖 )
post-measurement state is random; it is 𝜌𝑖 with probability 𝑝𝑖 , thus overall it is a mixed
state ∑𝑖 𝑝𝑖 𝜌𝑖 = ∑𝑖 𝑀𝑖 𝜌𝑀𝑖† . Note that the orthogonal measurement {𝑃𝑖 } (𝑃𝑖 ≽ 0, ∑𝑖 𝑃𝑖 = 𝐼) is
a special case because we can just take 𝑀𝑖 = 𝑃𝑖 , and note that 𝑃𝑖† = 𝑃𝑖 and 𝑃𝑖2 = 𝑃𝑖 .
The general quantum operation is quite similar. It turns out that for any quantum operation, one can
†
associate a collection of operators {𝐸𝑖 } in ∈ ℂ𝑀×𝑁 s.t. ∑𝑖 𝐸𝑖 𝐸𝑖 = 𝐼, and when operated on a
state 𝜌, the system is changed to ∑𝑖 𝐸𝑖 𝜌𝐸𝑖† . Note that the operator preserves the trace:
†
†
𝑡𝑟 (∑ 𝐸𝑖 𝜌𝐸𝑖 ) = 𝑡𝑟 (∑ 𝐸𝑖 𝐸𝑖 𝜌) = 𝑡𝑟(𝐼𝜌) = 𝑡𝑟(𝜌).
𝑖
𝑖
Exercise. What’s the operator-sum representation for the POVM measurement {𝑀𝑖 }? (hint: 𝐸𝑖 =
|𝑖⟩ ⊗ 𝑀𝑖 ).
1.2. CPTP maps
The second representation is by completely positive and trace preserving (CPTP) map.
Suppose the starting state is 𝜌 and an operation is Φ, then the ending state is Φ(𝜌).
Mathematically, Φ maps linear operators (like 𝜌) to linear operators (like Φ(𝜌)). Φ is
physically implementable if and only if Φ is a CPTP map.
Now we explain the term precisely. The “TP” part is easy to describe and understand: it
requires the map to preserve trace. Thus, it maps a quantum state 𝜌, which has trace 1, to
another quantum state which should also have trace 1.
The “CP” part is a bit more complicated. Recall that a square matrix 𝜌 corresponds to a
quantum state iff
𝑡𝑟(𝜌) = 1 and 𝜌 ≽ 0. While the trace part is handled by the above “TP”
property, the psd part is handled by the “CP” property. It is tempting to just require Φ to
preserve positivity: 𝜌 ≽ 0 ⇒ Φ(𝜌) ≽ 0. Such kind of Φ is called positive. Unfortunately
it’s not enough, because there are examples that when attaching a new system B, Φ ⊗ 𝐼𝐵 is
not positive any more. Physically, if we attach a new system B but operate only on the
original system A, the whole system should remain a valid quantum system, namely
(Φ ⊗ 𝐼𝐵 )𝜌𝐴𝐵 ≽ 0 should hold. For this to hold, Φ being positive is not enough. So we put
this further requirement in, and call a map Φ completely positive (CP) if Φ ⊗ 𝐼𝐵 is positive
for any B.
Now we know that
a map Φ is quantum admissible, i.e. implementable,
iff ∃{𝐸𝑖 } with ∑𝑖 𝐸𝑖† 𝐸𝑖 = 𝐼 s.t. ∀𝜌, Φ(𝜌) = ∑𝑖 𝐸𝑖 𝜌𝐸𝑖†
iff Φ is CPTP.
2. Distance measures of classical distributions
Suppose that we have two probability distributions 𝑝 and 𝑞 over a sample space 𝑋. Two standard
distance measures are


1
trace distance: 𝐷(𝑝, 𝑞) = 2 ∑𝑥∈𝑋|𝑝(𝑥) − 𝑞(𝑥)| = max(𝑝(𝑆) − 𝑞(𝑆))
𝑆⊆𝑋

∀𝑝, 𝑞, 𝐷(𝑝, 𝑞) ∈ [0,1]. 𝐷(𝑝, 𝑞) = 0 iff 𝑝 = 𝑞.

It’s a metric. Triangle inequality holds.
fidelity: 𝐹(𝑝, 𝑞) = ∑𝑥∈𝑋 √𝑝(𝑥)𝑞(𝑥)

∀𝑝, 𝑞, 𝐹(𝑝, 𝑞) ∈ [0,1]. 𝐹(𝑝, 𝑞) = 1 iff 𝑝 = 𝑞.

It’s not a metric. Triangle inequality doesn’t hold.
3. Distance measures of quantum states
We now extend the trace distance and fidelity to the quantum case. Suppose that 𝜌 and 𝜎 are two
quantum mixed states in the same space. Recall that for a real function 𝑓 and for a Hermitian 𝐻 =
∑𝑖 𝜆𝑖 |𝑢𝑖 ⟩⟨𝑢𝑖 | where 𝜆𝑖 ’s are the eigenvalues, one can define 𝑓(𝐻) = ∑𝑖 𝑓(𝜆𝑖 )|𝑢𝑖 ⟩⟨𝑢𝑖 |.

1
trace distance: 𝐷(𝜌, 𝜎) = 2 𝑡𝑟(|𝜌 − 𝜎|).

If 𝜌 and 𝜎 commute, i.e. 𝜌𝜎 = 𝜎𝜌, then 𝐷(𝜌, 𝜎) = 𝐷(𝜆(𝜌), 𝜆(𝜎)), where 𝜆(𝜌) is the
vector of eigenvalues of 𝜌, and similarly for 𝜆(𝜎).

Unitary operations don’t change trace distance: 𝐷(𝑈𝜌𝑈 † , 𝑈𝜎𝑈 † ) = 𝐷(𝜌, 𝜎).

𝐷(𝜌, 𝜎) = max 𝑡𝑟(𝑃(𝜌 − 𝜎)), where the maximum is over all projectors 𝑃.
𝑃

𝐷(𝜌, 𝜎) = max 𝐷(𝑝, 𝑞), where the maximum is over all POVM {𝐸𝑘 }, 𝑝 and 𝑞 are
{𝐸𝑘 }
distributions of outcomes obtained by applying {𝐸𝑘 } on 𝜌 and 𝜎, respectively.
This implies that the trace distance between 𝜌 and 𝜎 is the largest difference one can tell
by a measurement.

𝐷 is a metric. Triangle inequality holds.

Quantum operations don’t increase trace distance. (One cannot make two states more
distinguishable by operating on them.) 𝐷(Φ(𝜌), Φ(𝜎)) ≤ 𝐷(𝜌, 𝜎).


Strong convexity: 𝐷(∑𝑖 𝑝𝑖 𝜌𝑖 , ∑𝑖 𝑞𝑖 𝜎𝑖 ) ≤ 𝐷(𝑝, 𝑞) + ∑𝑖 𝑝𝑖 𝐷(𝜌𝑖 , 𝜎𝑖 ).
fidelity: 𝐹(𝜌, 𝜎) = 𝑡𝑟√√𝜌𝜎√𝜌

It’s symmetric: 𝐹(𝜌, 𝜎) = 𝐹(𝜎, 𝜌).

If 𝜌 and 𝜎 commute, then 𝐹(𝜌, 𝜎) = 𝐹(𝜆(𝜌), 𝜆(𝜎)).

𝐹(𝜌, 𝜎) = 1 iff 𝜌 = 𝜎.

𝐹(𝜌, 𝜎) = 0 iff 𝜌 and 𝜎 have orthogonal supports.

Special case of pure state(s): 𝐹(|𝜓⟩, 𝜌) = √⟨𝜓|𝜌|𝜓⟩, 𝐹(|𝜓⟩, |𝜙⟩) = |⟨𝜓|𝜙⟩|.

𝐹(𝑈𝜌𝑈 † , 𝑈𝜎𝑈 † ) = 𝐹(𝜌, 𝜎).

𝐹(𝜌, 𝜎) = max{|⟨𝜓|𝜙⟩|: |𝜓⟩, |𝜙⟩ purify 𝜌, 𝜎, respectively}

𝐹(𝜌, 𝜎) = max{|⟨𝜓|𝜙⟩|: |𝜓⟩ purifies 𝜌}, for any fixed purification |𝜙⟩ of 𝜎.

𝐹(𝜌, 𝜎) = min 𝐹(𝑝, 𝑞), where the maximum is over all POVM {𝐸𝑘 }, 𝑝 and 𝑞 are
{𝐸𝑘 }
distributions of outcomes obtained by applying {𝐸𝑘 } on 𝜌 and 𝜎, respectively.

𝐹(Φ(𝜌), Φ(𝜎)) ≥ 𝐹(𝜌, 𝜎).

𝐹(∑𝑖 𝑝𝑖 𝜌𝑖 , ∑𝑖 𝑞𝑖 𝜎𝑖 ) ≥ ∑𝑖 √𝑝𝑖 𝑞𝑖 𝐹(𝜌𝑖 , 𝜎𝑖 ).
A basic relation between the two distance measures is as follows.
1 − 𝐹(𝜌, 𝜎) ≤ 𝐷(𝜌, 𝜎) ≤ √1 − 𝐹(𝜌, 𝜎)2 .
Note
There are a couple of excellent references for quantum information. Part III of [NC00] is still
very good. [Wil13] is a new book with emphasis on quantum Shannon theory. [Wat11]
contains more other stuff and it is somewhat closer to computer science perspectives.
[KSV02] (Chapter 11 and 12) was one of the earliest introductions to channel distance by
diamond norm.
Reference
[KSV02] A. Yu. Kitaev, A. H. Shen , M. N. Vyalyi, Classical and Quantum Computation,
American Mathematical Society, 2002.
[NC00] Michael Nielsen and Isaac Chuang, Quantum Computation and Quantum Information,
Cambridge University Press, 2000.
[Wat11] John Watrous, Lecture notes for Theory of Quantum Information, Fall 2011.
[Wil13] Mark M. Wilde, Quantum Information Theory, Cambridge University Press, 2013